Remark 1.4 
From the point of view of the circle method (and the related delta method of [1]}, [2]}),
obtaining an asymptotic for (REF ) uniformly over \(H\approx x\)  (e.g. \(H\in [x^{1-\delta (x)}, x^{1+\delta (x)}]\) ) might only be tractable at present
for \(k=2\) , unconditionally (cf. Theorem 6 of [2]}),
and for \(k=3\) , conditionally (cf. [4]}).
Alternatively, using a “multiplicative” analog of the circle method along the lines of [5]}, [6]}, [7]}, one may in fact be able to obtain, for all \(k\) , an unconditional asymptotic (possibly involving many complicated “secondary” main terms) for \(H\)  close to \(x\) , but perhaps not for smaller \(H\) .
Nonetheless, for all \(k\) , we obtain an unconditional asymptotic for (REF ) uniformly over \(H\le x^{1-\delta (x)}\) ,
by over-extending the complicated “off-diagonal” contribution to (REF ) to a count of lower complexity;
see § for details.
It would certainly be interesting if one could handle a range of the form \(H\le x/(\log {x})^{O_k(1)}\) , even conditionally;
even for small \(k\)  (e.g. \(k=2\)  or \(k=3\) ), an analysis using [5]} or [1]}, [2]} could be enlightening in this regard.
We finally remark that a similar flavor counting question to (REF ) is studied in [11]} using the decoupling method.

